In my opinion, gameplay is arguably the most important aspect of any game, followed by sound, graphics, and finally story.
Gameplay is a given; if a game is not fun to play, than why should I bother playing it? That's why I'm drawn to Mario, Zelda, Metroid, and other Nintendo franchises. I also really enjoy games that are replayable, and I find that non-linear gameplay makes games more replayable. In addition, games should have comfortable, simple controls.
Sound is another aspect of games that I believe has to be done right. Sound effects, when done correctly, can really make a good first impression on a game. However, great, memorable soundtracks really create a lasting impression, which is another reason why I adore Nintendo franchises. I've hummed quite a few songs from the Mario, Zelda, and Metroid series, for instance. Not only that, but good music tracks add depth to an otherwise shallow game, setting the appropriate atmosphere.
Graphics are less important to me than gameplay or sound. While good graphics create a good first impression on a game and eye candy is always nice, I feel that after a while I stop noticing them. I tend to have lower standards for graphics because of this.
Story is even less important to me than graphics are. While I appreciate great stories, I don't think you should sacrifice gameplay or sound in the name of a better story. After all, video games are not books. I found that many of the games I like have simpler stories or barely a story at all, because the developers had focused more of their effort on gameplay or sound.